Abstract
A fractional Hardy–Sobolev inequality with a magnetic field is studied in the present paper. Under appropriate conditions, the achievement of the best constant of the fractional magnetic Hardy–Sobolev inequality is established.
1. Introduction
During the last decades, researchers paid more and more attention to the study of Sobolev inequalities and Hardy–Sobolev inequalities, including the fractional case and magnetic case, see e.g. [1–9].
It is well known that the sharp constant of the embedding is attained (see [10]), where is the Sobolev critical exponent and . That is,
is achieved by the so-called Aubin–Talenti instanton (cf. [11, 12]) defined by
where . Moreover, is a positive solution of in ,
and contains all positive solutions of in .
For the best constant of the Hardy–Sobolev inequality
where and is the Hardy–Sobolev critical exponent, it follows from [13] that is achieved by functions of the form
where . The function is a positive solution of in , and moreover,
For the fractional Sobolev inequality, consider the Hilbert space defined as Gagliardo seminorm
where , is the fractional Sobolev critical exponent and the norm
is induced by the scalar product
Here is a dimensional constant precisely given by
Define the best constant for the fractional Sobolev inequality as
From [14–16], we see that is achieved by , that is, . Normalizing by , we get that fulfills
and is a positive ground state solution of in (see Lemma 2.12 in [17]). Denote
where is a positive ground state solution of in and
Then, Lemma 2.12 in [17] yields that solves in . The fractional Hardy–Sobolev inequality
was considered in [18, 19], where is the fractional Hardy–Sobolev critical exponent. Marano and Mosconi [18] proved that is achieved by a optimizer , whose asymptotic behavior is
For the magnetic Hardy–Sobolev inequality, we regard the range of function as , that is, , is a magnetic vector potential. Setting , and
then and is the Hilbert space obtained as the closures of with respect to scaler product
where the bar denotes complex conjugation. Define
where is the Hardy–Sobolev critical exponent. Then, by Theorem 1.1 in [20], we see that if , then is attained by some if and only if , where curl is the usual curl operator for and the skew-symmetric matrix with entries for .
In our paper, we consider a fractional Hardy–Sobolev inequality with a magnetic field. To show our question, we first introduce the fractional magnetic Sobolev space , which is the completion of with respect to the so-called fractional magnetic Gagliardo seminorm given by
where is given by (10), , and is a magnetic vector potential. The scalar product in is
Although is a seminorm, by fractional magnetic Sobolev embeddings (see Lemma 3.5 in [21]), we can view as a norm in the space . As in Propositions 2.1 and 2.2 in [21], we can verify that is a Hilbert space. denotes the space of -integrable functions with respect to the measure , endowed with norm
The aim of the present paper is to investigate the following fractional magnetic Hardy–Sobolev inequality
where and is the fractional Hardy–Sobolev critical exponent. Problem (23) relates to the fractional magnetic Laplacian defined by
For and with mid-point prescription, the fractional magnetic Laplacian was studied in [21]. In particular, d’Avenia and Squassina [21] considered the operator
Obviously, (24) can be regarded as an extension of the above-mentioned operator involving mid-point prescription. The fractional magnetic Laplacian can also be defined by duality as
Denote
Then (23) can be characterized as:
Our main result is:
Theorem 1. If is a continuous function with locally bounded gradient, then is achieved by a nonzero element .
2. Proof of the Main Result
To prove Theorem 1, we need the following Lemma, which is obtained in [22].
Lemma 1 (Diamagnetic inequality). For any , we haveandwhich means .
By the fractional Hardy–Sobolev inequality (15) and Lemma 1, we have the following lemma.
Lemma 2 (Fractional magnetic Sobolev embeddings). The embeddingis continuous.
Proof of Theorem 1. Since the best constant of fractional Hardy–Sobolev inequality (15) is achievable, we only need to show thatIn fact, for any , there exists such thatSimilarly to Lemma 4.6 in [21], for any , consider the scalingSubstituting and , we derive thatand the following invariance of scaling:Noticing that for , we havewhere is compact support of . Obviously, a.e. in as . Since is locally bounded, for , we haveThen, there exists such that for ,DefineThen, for . Since
where may be different from line to line, we get that . By Lebesgue’s Dominated Convergence Theorem, we see that . Then, it follows from (33) thatwhich means that . The opposite inequality holds also because of Lemma 1. Thus, (32) holds, which completes the proof of Theorem 1.
Data Availability
The [inequality] data used to support the findings of this study have been deposited in the [references 21 and 22] repository.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Supported by NSFC (11701248), NSFLN (20180540028) and DSRFLNNU (BS2018L004).